rank correlation

Rank correlation

Ranks are the assignment of orders or priorities according to their status or importance. Karl Pearson’s correlation coefficient is especially useful when the data are quantitatively measured. There are some variables like beauty, knowledge, intelligence, honesty, etc. which cannot be measured quantitatively directly. These types of variables can be measured by assigning ranks or some sorts of ratings. Then the degree of association that exist between the two sets of rank is known as rank correlation. In such cases, there is a method by which the correlation between such characteristics can be studied. It is called rank correlation coefficient. This method was developed by the British Psychologist Charles Edward Sparkman in the year 1904.

Rank correlation is the degree of association between two variables when data are arranged in order or in ranks.

Sparkman Rank-correlation coefficient:

Sparkman developed a method of measuring rank correlation known as Sparkman rank correlation coefficient. It is generally denoted by r_s.The study of degree of relationship between different ranks or grades of two characteristics i.e. Sparkman’s rank correlation coefficient is defined by following formula:

r_s = 1 - \{\frac{6.∑d²}{n(n² – 1)}\}

where,

d= R₁ – R₂ = difference of rank between paired items

R₁= rank of first attribute

R₂= rank of second attribute

n= number of paired observations.

Properties of rank correlation coefficient

• The value of rank correlation coefficient ranges between -1 to 1.
• When R = +1 , then it denotes complete agreement in the order of ranks between the two attributes.
• When R= -1, then it indicates complete disagreement in the order of ranks.

For example,

r_s = 1 - \{\frac{6.∑d²}{n(n² – 1)}\}

r_s = 1 - \{\frac{6.0}{5(25 – 1)}\}

=1-0

=1

r_s = 1 - \{\frac{6.∑d²}{n(n² – 1)}\}

=1 - \{\frac{6.40}{5(25 – 1)}\}

=1-2

=-1

Note: the sum of differences of ranks between two variables should be zero i.e. ∑d=0.

There are three cases while calculating Sparkman’s rank correlation coefficient, they are as follows.

• When actual ranks are given
• When actual ranks are not given
• When the ranks are repeated

Case(I): when actual ranks are given

When the actual ranks are given then the following steps have to be followed;

1. Compute the difference of ranks (R₁ – R₂) which is denoted by d.
2. Compute d² to get ∑d²
3. Substitute these values in the formula of rank correlation coefficient i.e.

r_s = 1 - \{\frac{6.∑d²}{n(n² – 1)}\}

Example:

Ten industries of some state have been ranked as follows according to profit earned in and the working capital for that year:

Solution:

Here, ranks are given. By using the formula of Sparkman’s rank correlation coefficient, we can calculate rank correlation coefficient by following formula

r_s = 1 - \{\frac{6.∑d²}{n(n² – 1)}\}

computation of rank correlation coefficient

Rank correlation coefficient is given by,

r_s = 1 - \{\frac{6.∑d²}{n(n² – 1)}\}

= 1 - \{\frac{6.36}{10(100 – 1)}\}

=1- \{\frac{216}{990}\}

=1- 0.2182

=0.782

Case(II): when ranks are not given

when the actual data of the variables are given but not ranks, it will be necessary to assign the ranks. Ranks can be assigned by taking either from the highest value as 1 , second highest value as 2 and son on or the lowest value as 1, second lowest value as 2 and so on , then Sparkman’s rank correlation coefficient formula is used to compute rank correlation coefficient i.e.

r_s = 1 - \{\frac{6.∑d²}{n(n² – 1)}\}

example:

calculate Sparkman’s rank correlation coefficient between advertisement cost ( in thousand Rs.) and sales ( in lakh Rs.) from the following data.

Solution:

In this problem, ranks are not given. We have to assign ranks starting from the smallest or from the largest value for both cases. Let us rank starting from the largest value. Let X and Y denote the advertisement cost ( in thousands Rs.) and sales ( in lakhs Rs.) respectively.

Table for calculation of rank correlation coefficient

Now ,

r_s = 1 - \{\frac{6.∑d²}{n(n² – 1)}\}

= 1 - \{\frac{6.30}{10(100 – 1)}\}

=0.82

Case(III): when the ranks are repeated(tied)

For some cases when more than one items have r\the same value within a series, a common rank is given to all such repeated items. These common ranks are the arithmetic mean of the ranks which these items would have got if they were different from each other and the next item will get the rank next to them. For example, if third and fourth items have the same value, then the rank for each of them third and fourth items 3.5 and the rank of the next item is 5. For this case, the general formula (as discussed above) for calculating rank correlation coefficient is adjusted by adding a tie correlation factor \{\frac{m(m²-1)}{12}\} to ∑d². Then the formula, for calculating Sparkman rank correlation coefficient, when more than one items have the same value, is as follows.

r_s=1- \{\frac{6[∑d² + \{\frac{m₁(m₁²-1)}{12}\} + \{\frac{m₂(m₂²-1)}{12}\} +………}{n(n²-1}\}

Where, m₁,m₂,……etc. be the number of times that an item is repeated.

Example:

An examination of 10 applicant was taken by a firm. From the marks obtained by the applicants in statistics and accountancy papers. Calculate the rank correlation coefficient.

Solution:

Let R₁ denote the rank of marks in statistics and R₂ denote the rank of marks in accountancy.

Computation of rank correlation coefficient

Here, n= 10, m₁= 3, m₂=2, m₃=2

Rank correlation coefficient

r_s=1- \{\frac{6[∑d² + \{\frac{m₁(m₁²-1)}{12}\} + \{\frac{m₂(m₂²-1)}{12}\} +………}{n(n²-1}\}

=1- \{\frac{6[83 + \{\frac{3(3²-1)}{12}\} + \{\frac{2(2²-1)}{12}\} + \{\frac{3(3²-1)}{12}\} }{10(10²-1}\}

=1- \{\frac{6*30}{990}\}

=1-0.1818

=0.82

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Dhakal Bashanta (2014).Business Statistics,Buddha academic publisher

Sthapit, Azaya Bikram(2006),Business Statistics,Asmita publication